Definition 3

Let f be a mapping of a set X into a topological space Y, and let $\mathfrak{F}$ be a filter on X. A point $y \in Y$ is said to be a limit point (or simply a limit) (resp. cluster point) of f with respect to the filter $\mathfrak{F}$ if y is a limit point (resp. cluster point) of the filter base $f(\mathfrak{F})$.

The relation "y is a limit of f with respect to the filter $\mathfrak{F}$" is written $\lim_{\mathfrak{F}} f = y$, or $\lim_{x, \mathfrak{F}} f(x) = y$, or $\lim_x f(x) = y$ if there is no risk of confusion.

From Definition 3 and Propositions 1 (no. 1) and 3 (no. 2) we deduce the following criteria: